A Differential Equation–Based Model of Toxin-Mediated Feedback in Bioaugmented Water Remediation with Optimal Control

Abstract

Bioaugmented remediation uses pollutant-degrading bacteria to treat contaminated water, a low-cost alternative to chemical treatment. But many pollutants, like toluene, are toxic to the very bacteria meant to break them down. In species such as Pseudomonas putida, poisoned cells lyse, releasing both unmetabolized pollutant and a secondary toxin back into the system. We build a five-compartment ODE model coupling susceptible, intoxicated, and resistant degrader populations to pollutant and toxin levels, and prove that solutions stay nonnegative and bounded. Using the next-generation method, we derive a basic reproduction number that splits cleanly into two feedback pathways: pollutant recycling and toxin release. We show recycling alone is subcritical and can only initiate contamination, while the toxin pathway is what keeps it going; We prove the existence and uniqueness of an endemic equilibrium whenever the basic reproduction number is greater than one. Ablation experiments show that cutting the toxin loop clears the system, while cutting the pollutant loop only reduces total burden by 15%. The model exhibits persistent oscillations under baseline parameters and no dosing, meaning contamination keeps recurring without intervention. Applying Pontryagin's Maximum Principle with a forward-backward sweep algorithm, we identify an optimal early pulse dosing strategy for a protective agent, reducing total remediation cost by 67.2% compared with no intervention and 34.8% compared with continuous maximum dosing. These findings identify toxin-mediated feedback as the dominant driver of persistent contamination and provide quantitative guidance for designing efficient bioaugmentation strategies that support UN Sustainable Development Goal 6: Clean Water and Sanitation.